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Full text of "Treatise On Analysis Vol-Ii"

13    EXTENSIONS OF HERMITIAN OPERATORS       429

or unbounded) open interval in R, and 3f by the subspace &([) of indefinitely
differentiable complex-valued functions on I with compact support. It is
clear that the graph of D is not closed: for example, it contains the pairs
(X D#) where x is of class C1 but not of class C2, and Supp(x) is compact
(17.1.2).

(15.13.3)    Let H be a hermitian operator. Since dom(#*) is dense in E, the
closure H** of H exists, and since its graph is the closure of the graph of
H (15.12.4), it follows from (15.13.1.1) by continuity that H** is hermitian. We
may therefore restrict our considerations to closed hermitian operators.

(15.13.4)    (i)   Let H be a closed hermitian operator on E. For each real
number a ^ 0, the closed operator H -+ ail: XH- H  x -f odx is an injective
mapping ofdom(H) into E, whose image Fa is a closed subspace ofE, and the
operator (H + a/7)"1 (15.12.5) is continuous on Fa.

(ii)   The linear mapping

(15.13.4.1)                 V : x^H-iTKH+iir1 ' x

of Pi into E is an isometry ofV^ onto the closed subspace V(F1) = F_A. The
mapping I  V: F1 = dom(F) -+Eisa bijection 0/dom(F) onto dom(H), and
we have

(15.13.4.2)                     H-y = i(I+ V)(I- V)~l 'y

for all y e dom(/f).

(iii) Conversely, let F be a closed subspace ofE, and U an isometry off
onto a subspace o/E, such that the image GofF = dom(U) under / U is
dense in E. Then the operator I  U is a bijection ofF onto G; if, for each
y e G, we put

(15.13.4.3)                      H-y = i(I+ U)(I- U)

then H is a closed hermitian operator, with dom(H) = G; moreover, the
operator V defined in (15.13.4.1) is equal to U.

By virtue of (15.13.1.1) we have

(15.13.4.4)         \\H-ynging to ^